If $a^{x} = b^{y} = c^{z}$ and $x,y,z$ are in G.P., where $a,b,c,x,y$, z are positive real numbers n

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If $a^{x} = b^{y} = c^{z}$ and $x,y,z$ are in G.P., where $a,b,c,x,y$, z are positive real numbers not equal to unity, then :

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Sol. [Key Concept: $a^{x} = b^{y} = c^{z}$ ⇒ take logs; combine with $a,b,c$ in G.P. to relate the logarithms.]$$xlna = ylnb = zlnc$$$$\begin{matrix} \Rightarrow & \frac{lna}{ln	ext{ }b} = \frac{y}{x} = \frac{z}{y} = \frac{lnb}{lnc} \ \Rightarrow & \log_{b}a = \log_{c}	ext{ }b \end{matrix}$$

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